Originally from the Metrology Synthesis thread, but I thought it deserved its own thread:

wendy.krieger @ Jan 4 2014, 08:05 AM wrote:Preferred Numbers

If you lay out the 2-3 numbers (ie \( 2^x 3^y \) in a grid by x,y, you get these numbers.Â If you now draw lines through mantissa-10; eg 1,12,144, etc and 2,24,288, etc, you get the 'awayness' lines.Â This is a measure of how far away it is from powers of 12.

The preferred numbers in the 'FS(n)' series is then those lines that are up to 2n away from the power-of-12 line.Â So for fs(2), you get 9 numbers 1, 14, 16, 2, 3, 4, 6, 8, 9.Â

If you instead take the units of weight, using, say cof as the base measure, and tgm as the other limit, you have tgm lying 6 awayness from cof.Â This means that from a cof prospective, there are six lines between cof and tgm (1, 6, 3, 16, 9, 46, 23), and from tgm, there are six (1, 2, 4, 8, 14, 28, 54).Â So if you keep your units inside these limits then it has an easy conversion from the DD to cof or tgm.Â

So if you pick units between the corresponding tgm and cof awayness lines, you end up with a system of units that serves general needs, but we have nearly all of the derived units also fall in this range.

Some evidently don't, eg PSI = 2;80 foot-head, or 80; corn-heads.Â But most do.

As of 1202/03/01[z]=2018/03/01[d] I use:
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[i]More of the exchange from the [url=http://z13.invisionfree.com/DozensOnline/index.php?showtopic=1108&view=findpost&p=22113630]Metrology Synthesis[/url] thread:[/i]
[quote="Kodegadulo @ Jan 5 2014, 04:55 PM"]This is a great idea. It would help to actually see it. But if you don't mind, there are twice as many 2's than 3's in dozenal, and these browsers are finicky about really wide tables, so rather than do it as x columns of 2's and y rows of 3's, I'd like to lay this out as n rows of 2's and m columns of 3's:
...
I wish I knew how icarus did the background colors in his number base tables. It would be nice to colorize this table to show "awayness" and/or FS(n) level. [/quote]
[quote="wendy.krieger @ Jan 6 2014, 07:05 AM"]The code to colour individual cells is bgcolor=red &c.[/quote]
[quote="Kodegadulo @ Jan 7 2014, 04:03 AM"]Thanks!
Here's that table, colorized. Wendy I'm not sure I understand your definition of "awayness", but Dan's concept of [url=http://z13.invisionfree.com/DozensOnline/index.php?showtopic=264&view=findpost&p=3738597][b]regularity order[/b][/url] provided what I think was a good definition of "awayness": It indicates the number of steps away from an unqual power a given number is.
[/quote]

As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
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wendy.krieger @ Jan 7 2014, 10:24 AM wrote:Awayness.

[decimal] It is just a kind of if you write, in place of \( 2^m 3^n \), you use \( 12^a 2^b \) or \( 18^a 6^b \) or \( 72^a 12^b \), the measure of 'b' is the awayness in those bases. All bases of two prime divisors are like this, but proper powers are multi-layered. For example, base 36 is actually \( 6^a 2^b \), because 36 is a power of six anyway.

Doing something like FS() is alright, but if you are going to do something between COF and TGM, then you don't really want to include both +7 and -7 in either scale. There's only seven lines in mass between COF and TGM. So even though the ratios between COF and TGM lie in FS(3), you don't need all twelve lines between the two. The middle stripe, and six on one side suffices.

When the base has three prime divisors, things change. Instead of having 2^n vs 3^m, you first suppose that numbers have already been lined up mantissa of the base, so that 2 represents all \( 2.b^n \). You then set the regulars out so that up and down come powers of 2, and its inverses (60, 30, 15, 760, 390, 1V5 &c.)

Across you have 10 and 12 opposite each other. The FS lines are not just parallel bars (or dots), but whole rings of increasing size. However, the FS() scale is not the thing that renders calculations small. It's how many steps you take to jump from one number to another. It's like in your graph, keeping to near one stripe of green, against jumping between the green below the powers of 12, to those above it.

In base 18, the prime means add nine to the next digit, so '8 means '17' etc. Base 72 is in six/dozen, the number after it shows the fs() point. Base 56 is eight sevens, has the tightest opposition in any base < 100, and base 100 is shown as far as fs(3).

If you look at the twelfty one, it looks like a box in prospective. It's no accident here. That's what fs(n) does. It's also the basis of all those nifty formulae i made for icarus. i mean, volume Ã· volume gives number.

As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
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Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
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(Links to these and other useful topics are in my index post;
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[i]Originally from the [url=http://z13.invisionfree.com/DozensOnline/index.php?showtopic=1108&view=findpost&p=22113678]Metrology Synthesis[/url] thread:[/i]
[quote="Kodegadulo @ Jan 7 2014, 12:50 PM"]Very difficult to decipher all this...
Let me see if I can even puzzle out your notation:
For base 18[sub]d[/sub], you label it as decimal, which makes it sound like you're using a non-multistaff large base with decimal sub-base [18[sub]d[/sub]] and superdigits 00, 01, 02, 03, 04, 05, 06, 07, 08, 09, 10, 11, 12, 13, 14, 15, 16, 17. But what you really have here is a true binary/ennary multistaff base, which can be labelled [29] or [be] (depending on taste), with superdigits 00, 01, 02, 03, 04, 05, 06, 07, 08, 10, 11, 12, 13, 14, 15, 16, 17, 18. But you've created a shorthand notation for those digits where a 0 in the binary subdigit is omitted, and a 1 is encoded as a prime (or apostrophe). Base 18[sub]d[/sub] might also be done as a simple base I, if you can stomach using digits 0 1 2 3 4 5 6 7 8 9 A B C D E F G H:
...
Except that for that last example you've got '9 = 19[sub]29[/sub] = 18[sub]d[/sub] = I[sub]I[/sub] -- but this seems to be an error, because I is not a digit of base [ I ], any more than dek would be a digit of decimal base. (I'd have colorized the bad digit in the base [18[sub]d[/sub]] version exception that I can't do that in Tex/Mathjax.)
...
[/quote]
[quote="Kodegadulo @ Jan 8 2014, 01:01 AM"]Actually, Wendy, if you're really using multistaff binary/enneary, then [i]all[/i] those nines are wrong, because 9 cannot be a digit in base 9. You'd have to express it as '0, and I should have used superdigit 10. Didn't catch that the first time. Here's what I should have written (I'm switching to prefix notation because it's just easier):
(Just for fun, I also threw in base dozen-six as a non-multistaff large base with dozenal sub-base.)
If you're not using a multistaff 29| base, then I dunno what to call it. It would be a redundant notation because you'd have two ways to express the same digit: 9 = '0.[/quote]

As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
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Awayness was originally used for something entirely different. Specifically it set a relation between bases. For example, 18 and 24 are at fives with each other, and 18 and 48 at sevens.

The relation of awayness is perfectly recriprocal. The awayness of 18 in base 48 is equal to the awayness of 48 in base 18, both being 7. This means, that in either base, the other stands not just on a line seven removed from the base, but one that contains seventh powers. Some product of 18 and 48, not involving seventh powers, gives a seventh power. For example, b18 2'3000 is the seventh power of 6, and in b48, one has 18.0.0.0.0.0 as the seventh power of 24.

This means, that if the index of a prime (ie the p-1 divided by the period), is a multiple of 7 in one, it is too in the other. It covers prime powers too, but the period is p^{ n-1}(p-1).

The relation of bases is used as a guide for factorising algebraic roots of \( b^n -1 \). Fermat's little theorm tells you that Ax consists of primes of the form xy+1, and usually at most one repeater. Knowing the relation between two numbers helps things along.

In dozenal, the bases 10 and 80 are at threes. The prime 81 is then has an index that is a multiple of 3 , because for dividing A2, it is 40. So 81 has an index that is even (quadratic recriprocal), and a multiple of 3, because it has the same in a base 10 is at threes with. So the maximum period is 80/2/3=14. But since the factors of A1, A2, A4 and A8 are known, and do not hold 81, we note that 81 divides a14, and thus has a 14 place period in dozenal.

Likewise, with 16, which is at fives with 20 and at sevens with 40, one sees that 21 is 5*5 and 41 is 7*7, that for 5 dividing its own period in 20, and 7 in 40, then 5 and 7 must divide their own periods in 16 too.

Wendy, perhaps you could demonstrate that with a few examples. I'm sure there would be math fans interested.

As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
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The rule of lines is to single out the close regulars. I shall use decimal here, but it works in all bases.

A single place gives the dividers of 10, to wit, 1, 2, 5. Each of thses are multiples of dimes, divide the dollar. As multiples of the dollar, divide the eagle, and so forth. We could write these as 1|0|. That is, for being multiples of one place, are divisors of the next.

One can enclose any number of places in this way. 1|000| gives those that are multiples of mm but divide the metre. There are certain mantissa that are new here, not found in eg |00|. The order of a regular is then the minimal number of 0 fall between the bars. These might be arranged by mantissa.

A factorseries is then all those less than order n.

The regulars for base 18 are (opposite powers of 3)

1 6 2 C 4 16 8 2C G 56

These are numbers of the form 18^x 6^y, and if the number itself is not a proper power (like C0 is 6^3), then the relation between them is at 3's with. So it means that if two numbers are at x's with each other, then (p-1)/period are the same.

For example, with 12 and 18, the relation is threes, and you get this. Note that 10*8 is also at threes with 10.

The thing to note here is that the period of one base divides p-1 to give a multiple of three, the other one does too. For example, the period of 157 in B12 gives 156/3, = 52, the base 18 gives 156/156 = 1. Note that with 19, B12 gives 3, and B18 gives 9.

wendy.krieger @ Oct 18 2015, 06:38 AM wrote:These are numbers of the form 18^xÂ 6^y, and if the number itself is not a proper power (like C0 is 6^3),

What do you mean by, not a "proper" power? Proper power of what? Looks like a perfectly proper power of 6 to me. It's also a perfectly fine example of the formula 18^x 6^y you just cited, simply substitute x=0 and y=3. You've got people thoroughly confused already.

then the relation between them is at 3's with.Â Â So it means that if two numbers are at x's with each other, then (p-1)/period are the same.

You know, in mathematical discourse, most people would introduce a new and unfamiliar term for a new kind of relation by defining the general case first, before using an example of it. The relation is terms of some x, so where does the p come from?

ForÂ example, with 12 and 18, the relation is threes, and you get this.Â Note that 10*8 is also at threes with 10.Â

Totally lost. Which of these numbers are supposed to be the bases, which the primes, and which the periods? I see a bunch of prime numbers, but is 9 supposed to be a prime? It's composite. This table does not line up nicely at all when viewed on a cell phone, nor when looking at it in the Quote editor on a desktop. Is it really too much to ask you to format something like this with a <table> tag in a [ dohtml ] block, and put some actual labels on your columns and rows?

The thing to note here is that the period of one base divides p-1 to give a multipleÂ of three, the other one does too.Â For example, the period of 157 in B12 gives 156/3, = 52, the base 18Â gives 156/156 =Â 1.Â Note that with 19, B12 gives 3, and B18 gives 9.

At first I thought B12 and B18 were supposed to be based numbers in base 18, then I realized it was just your shorthand for saying "base 12" and "base 18". Come on now, is it really such a chore to type out the word "base"?

As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

"Proper" is a pretty ordinary mathematical term, which means not including the unit case. For example, 60 is a proper divisor of 120, but 120 isn't.

A number like 12^1, is a power, but not a proper power, since it is the first power. Something like 6^6/18^3 is a proper power, since it is equal to a power of a smaller number: 2^3. Note that eg 24*18, given as 6^5/18^2, is 432, is not a proper power, since no number raised to something greater than the first, gives 432.

The relation of at x's with, means that a^m b^n = c^x, where the gcd(x,m)=gcd(x,n)=1. This means that gcd(i(m), x) divides gcd(i(n),x) for all primes, where i(n) is (p-1)/period(p,n) and period(p,n) is the period-length of p in base n.

The table showing primes 6n+1, the primes are in the columns, and the bases are in the rows, so the top half shows bases 12 and 18, for which 12*18 = 6^3, and the second shows 10 and 80, where 10Â²*80 = 20Â³.

Note the presence of 9 in this list. The sevenites are included in this process as well, so if two bases are at a relation of 'fives' (such as 18 and 24) or 'sevens' (as 18 and 48 are), and 5 or 7 is a sevenite in 18, it is in bases 24 and 48 as well.

wendy.krieger @ Oct 18 2015, 11:08 AM wrote:"Proper" is a pretty ordinary mathematical term, which means not including the unit case. For example, 60 is a proper divisor of 120, but 120 isn't.

A number like 12^1, is a power, but not a proper power, since it is the first power. Something like 6^6/18^3 is a proper power, since it is equal to a power of a smaller number: 2^3. Note that eg 24*18, given as 6^5/18^2, is 432, is not a proper power, since no number raised to something greater than the first, gives 432.

Don't be patronizing Wendy. Even if we understand what "proper" means in principle, long experience has shown that you tend to mangle terms and leave out critical bits of context. For instance, while 2^3 may be a proper power of 2 it is not a proper power of 8. If you meant to say that the numbers need to be proper powers of _some_ integer, you could have said that, or better yet you could have said "x must be a proper power of some integer, i.e. there must be some integers m,n both greater than 1 such that x=m^n". It's best to provide both the English description and the precise mathematical specification.

The relation of at x's with, means that a^m b^n = c^x, where the gcd(x,m)=gcd(x,n)=1. This means that gcd(i(m), x) divides gcd(i(n),x) for all primes, where i(n) is (p-1)/period(p,n) and period(p,n) is the period-length of p in base n.

You see, that's more like it.

The table showing primes 6n+1, the primes are in the columns, and the bases are in the rows, so the top half shows bases 12 and 18, for which 12*18 = 6^3, and the second shows 10 and 80, where 10Â²*80 = 20Â³.

Note the presence of 9 in this list.

I already have noted it. Why is it among a collection of "primes", when it is composite? There also seems to be some misalignment, so I couldn't tell if you meant the 9 to be a base or not. Really, if you can't be bothered to do some formatting to make a clear table, don't be surprised if your observations get little currency.

As of 1202/03/01[z]=2018/03/01[d] I use:
ten,eleven = ↊↋, ᘔƐ, ӾƐ, XE or AB.
Base-neutral base annotations
Systematic Dozenal Nomenclature
Primel Metrology
Western encoding (not by choice)
Greasemonkey + Mathjax + PrimelDozenator
(Links to these and other useful topics are in my index post;
click on my user name and go to my "Website" link)

The terminology I use for bases derives from tablesin the CRC Handbook of Mathematics, from photocopies of pages out of L E Dickson History of Mathematics, and perhaps Yates 'Repunits', along with Wells 'Dictionary of interesting and curious numbers'.

This is woefully inadequate coverage, so i invented edtra terms.

Suppose that in base b, the period of a prime p is n. For ordonary 'class 1' primes, n divides p-1. The discussion is not around the period but the number of co-periods, that is how many different loops there are. This is the reduced index.

CRC mentions a primiive root, the smallest number g for which if g^n = 1, then p-1 divides n. The primitive root needs to be ajusted occasionally for sevenites, that is, if p^2 divides g^p - g then one moves onto the next These instances i call sevenites in g,

g^i passes through every number from 1 to p-1. One can tabulate i for each prime 2-19 etc, these add to get value i, the number of loops is gcd(i, p-1), and the period is (p-1)/gcd(p-1, i).

One calls base b/a as b^n - a^n, and this offers "algebraic roots", one different for each N. You can write these in algebraic form, or in base form.

Primes 'saying' x may divide Ax. if p divides Aj, then p divides pApj, which means that p is saying pj too, and the entry might be used to handle sevenites at that order.

If p and q are at some relation c's, then there is some p^a q^b = r^c, where gcd(a,c)=gcd(b,c)=1. Two numbers at the relation of c's, the loop-lengths of two primes a, b, follow the pevious relation.

Note the relation of being at c's is transitive, and that if one of the two is a proper power, then the relation holds only for the complement.

For example, 8 and 18 are sixes with each other. But base18[ 8 is a proper cube, so the actual relation is at twos. Likewise 80 is a proper square, and the relation is a cube, and 800 is a simple number (dec 2592), the relation is sixes.]

The proper title of the table i give is p saying 3, and includes 9, for which the euler totient is the same as 7, and like 7, has a primitive root.